The paper constructs compact λ-torus in Euclidean spaces.
problem Creating compact embedded λ-hypersurfaces with torus topology. method Constructing compact embedded λ-hypersurfaces with specific topology. result Compact embedded λ-torus in Euclidean spaces Rn+1 constructed. Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
Complete classification of rod complements in 3-torus using topology.
problem Classifying rod complements in the 3-torus.
method Topological arguments.
result Complete classification of all rod complements in the 3-torus.
Abstract: Topological quantum field theory connects graph evaluations to polynomial identities.
problem Graph evaluations in topological quantum field theory.
method Relates SO(3) topological quantum field theory trace evaluations to topological Tutte polynomial evaluations.
result Generalizes the Tutte golden identity for graphs on the torus.
Sharp signature bound for 3-strand torus knots proved.
problem Determining the topological 4-genus of 3-strand torus knots.
method Using McCoy's twisting method and improving upper bounds.
result Signature bound is sharp for 3-strand knots and off by at most 1 for 4- and 6-strand knots.
We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.
New approach simplifies topological T-duality for torus bundles.
problem Global assumptions on H-flux in T-duality.
method Introducing a new 'Thom class' formulation.
result Easier and more transparent proofs of T-duality.
Paper proves mirror symmetry for conifold transitions of torus knots.
problem Relating open Gromov-Witten invariants to topological recursion.
method Uses topological recursion on spectral curves.
result Proves mirror symmetry conjecture for conifold transitions.
A new method analyzes topological B-model on a torus using doubled geometry.
problem Reproduce derived category of coherent sheaves on a torus.
method Double field theory and T-duality in doubled geometry framework.
result Correctly computes BRST cohomology and derived Hom-spaces of line bundles.
297 unique triangulations found for a punctured torus.
problem Finding all unique triangulations of a once-punctured torus.
method Hand enumeration of irreducible triangulations.
result Exactly 297 non-isomorphic triangulations exist.
We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…
New invariant defined for unoriented knots, proving no factorization through topological concordance.
problem Defining and proving properties of unoriented slice-torus invariants.
method Introducing and proving properties of unoriented slice-torus invariants.
result Unoriented slice-torus invariants do not factor through the topological concordance group.
The paper studies Lefschetz pencils on four-torus, proving their uniqueness and constructing examples.
problem Understanding the topology of Lefschetz pencils on the four-torus.
method Using moduli spaces of polarized abelian surfaces, the authors prove the uniqueness of pencils and construct examples.
result Holomorphic Lefschetz pencils on the four-torus are uniquely determined by their genus and divisibility.
Study geodesics in 3-torus, determining complements' topology.
problem Understanding the topology of geodesic complements in 3-torus.
method Analyzes the orbit of direction vectors under PSL3(Z) action and uses Farey graph distances. result Determines homeomorphism type of geodesic complements in 3-torus.
New insights into non-torus links using topological vertices.
problem Understanding non-torus links through topological vertices.
method Tangle calculus and topological string considerations.
result Explicit description of a non-torus link L8n8. We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficien…
Study torus knots in lens spaces using Gromov-Witten invariants and topological recursion.
problem Computing Gromov-Witten invariants for torus knots in lens spaces.
method Construct Lagrangian sub-manifolds and relate to topological recursion.
result Verify a conjecture in lens space for Gromov-Witten invariants.
Efficient algorithms for WRT invariants of torus bundles using algebraic structures.
problem Computing topological invariants of 3-manifolds is generally intractable.
method Embedding skein algebra into symmetric subalgebra at roots of unity for polynomial-time classical computation and using quantum algorithms for exponential space advantage.
result Polynomial-time classical computation and quantum algorithms for WRT invariants of torus bundles.
Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free t…
Study the complexity of horizontality in 4-torus vector bundles.
problem Classify topological holonomy groups in SO(3).
method Analyze twistor spaces and oriented vector bundles over 2-torus.
result Discover many topological holonomy groups in SO(3) with noncommutative pairs.
We study the invariant of knots in lens spaces defined from quantum Chern-Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the paper, we propose a B-model topological string theory description of torus knots in…
Topological recursion recovers a specific partition function for colored knots.
problem Recovering the extended Ooguri-Vafa partition function for colored HOMFLY-PT polynomials of torus knots.
method Applying topological recursion to the spectral curve of colored HOMFLY-PT polynomials of torus knots.
result Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.
Study of rod packings in 3-torus using 3-manifold geometry.
problem Understanding crystal structures in crystallography through rod packings in 3-torus.
method Use of 3-manifold geometry and topology to analyze complements of rod packings.
result Find families of complements that are hyperbolic and Seifert fibred.
The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under C0 perturbations. method Study of topological entropy on Riemannian metrics with C0 topology. result Metrics with contractible closed geodesics have robust entropy.
Ancient solutions to Ricci flow on torus bundles have additional symmetries.
problem Understanding collapsed ancient solutions to the Ricci flow on compact manifolds.
method Algebraic and tameness assumptions on collapsing directions to prove additional torus symmetries.
result Ancient solutions to the Ricci flow on torus bundles converge to an Einstein metric on the base.
This paper extends knot invariants using instantons to study torus knot groups.
problem Understanding the topology of knots and their representations.
method Generalization of equivariant singular instanton Floer theory.
result Irreducible singular instanton homology of torus knots for rational holonomy parameters are Z/4-graded abelian groups. Study rigidifies torus bundles under first Betti number constraints.
problem Understanding the structure of torus fibrations under first Betti number restrictions.
method Established rigidity results and necessary/sufficient conditions for topological splitting.
result Classification of torus bundles under specific Betti number constraints.
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should …
In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.
Study describes Morse flows on a torus with up to six singular points.
problem Understanding the structure of Morse flows on a torus with a hole.
method Used separatrix diagrams to describe topological structures and saddle-node bifurcations.
result Identified all possible topological structures of Morse flows with at most six singular points.
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
We point out that a 4-dimensional topological manifold with an Alexandrov metric (of curvature bounded below) and with an effective, isometric action of the circle or the 2-torus is locally smooth. This observation implies that the topological and equivariant classifications of compact, simply connected Riemannian 4-ma…
Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.
problem Character variety and topological invariants of Dehn fillings.
method Extension problem for quaternion Azumaya algebras.
result New phenomena in extension problem not seen in knot complements.
We prove the existence of essential loops in the space of contact structures on torus bundles over the circle.
Study torus orbifolds with two fixed points and their cohomology.
problem Understanding the topological and cohomological properties of torus orbifolds with two fixed points.
method Analyze the equivariant topological type and use results from [DKS] to compute integral equivariant cohomology.
result Results on generators and relations for the cohomology of torus orbifolds with two fixed points.
Study calculates Kashaev invariants for twice-iterated torus knots.
problem Calculating Kashaev invariants for complex knots.
method Asymptotic analysis and topological interpretations.
result Obtained formulas linking invariants to Chern-Simons and Reidemeister torsion.
Polynomial algorithm for multiplication on one-hole torus skein algebra.
problem Complexity of multiplicative structure in skein algebra.
method Provided a polynomial algorithm for one-hole torus.
result Closed form formulas for multiplication of curves with low crossing number.
Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying $rank(T) =…
Computes structure set of manifolds homotopy to torus bundles over lens spaces.
problem Computing the structure set of manifolds homotopy equivalent to certain torus bundles.
method Computes topological simple structure set using surgery invariants.
result First computation of structure set for manifolds with non-trivial fundamental group.
Study centers of mapping-torus groups to define knot and mapping class invariants.
problem Understanding the center of mapping-torus groups.
method Determine the center of meta-nilpotent quotients of mapping-torus groups.
result Introduce two invariants of knots and mapping classes as quadratic forms.
Research classifies L-space knots using Floer simple manifolds.
problem Understanding L-space fillings and their topological significance.
method Topological characterisation of Floer simple manifolds and classification of L-space knots.
result Partial classification of L-space twisted torus knots.
This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology …
Builds torus fibrations over singular manifolds with specific features.
problem Creating torus fibrations over manifolds with singularities.
method Constructs a topological space X as a torus fibration over an integral affine manifold B with singularities. result The fibration has a discriminant in codimension 2.
New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.
The paper uses twisting operations to bound knot genus.
problem Bounding the topological slice genus of knots.
method Develops null-homologous twisting operations to study algebraic genus.
result New upper bounds on algebraic genera of torus and satellite knots.
Authors classify 3D locally standard T-pseudomanifolds under weaker conditions.
problem Classifying equivariant homeomorphism types of 3D locally standard T-pseudomanifolds.
method Introduced and classified by characteristic data under homotopy equivalence condition.
result Condition for classification can be removed when dimension is at most three.
Elementary proof shows no specific torus to sphere cover with certain branching points.
problem Existence of specific branched covers between torus and sphere.
method Elementary topological proof using properties of the torus.
result No such branched cover exists with specified branching points.
Study surgeries between lens spaces using Heegaard Floer d-invariant.
problem Classify surgeries between lens spaces of type L(n,1). method Developed a d-invariant surgery formula for L-space knots. result Classified distance one surgeries between lens spaces of the form L(n,1).